Abstract
We present a new smoothing method based on a logarithm-exponential function for mathematical program with complementarity constraints (MPCC). Different from the existing smoothing methods available in the literature, we construct an approximate smooth problem of MPCC by partly smoothing the complementarity constraints. With this new method, it is proved that the Mangasarian-Fromovitz constraint qualification holds for the approximate smooth problem. Convergence of the approximate solution sequence, generated by solving a series of smooth perturbed subproblems, is investigated. Under the weaker constraint qualification MPCC-Cone-Continuity Property, it is proved that any accumulation point of the approximate solution sequence is a M-stationary point of the original MPCC. Preliminary numerical results indicate that the developed algorithm based on the partly smoothing method is efficient, particularly in comparison with the other similar ones.
Highlights
Consider the following mathematical program with complementarity constraints (MPCC): min f (x) s.t. g (x) ≤ 0, h (x) = 0, (1) G (x) ≥ 0, H (x) ≥ 0,G (x)T H (x) = 0, where f : Rn → R, g : Rn → Rm, h : Rn → Rp, and G, H : Rn → Rl are all continuously differentiable functions
We present a new smoothing method based on a logarithm-exponential function for mathematical program with complementarity constraints (MPCC)
Some specific approaches have been proposed for solving MPCC (1), such as the sequential quadratic programming approach in [3,4,5,6,7,8], the interior point methods in [9, 10], the penalty approach in [11,12,13], the lifting method in [14], the relaxation approach in [15,16,17,18,19,20,21,22], and the smoothing methods in [23,24,25,26,27,28,29,30]
Summary
The smoothing method is one of the most popular approaches, which uses a smoothing function to approximate the complementarity constraints in (1). Mathematical Problems in Engineering to a stationary point (or an optimal solution) of the original MPCC. In [31], a partially smoothing Jacobian method is proposed for solving nonlinear complementarity problems with P0 function. Under a weaker constraint qualification, called the MPCCCone-Continuity Property (MPCC-CCP), we can prove theoretically that any accumulation point of the approximate solution sequence is M-stationary to the original MPCC. We first review some concepts of nonlinear programming and MPCC; we present a new smoothing method of Problem (1). For a given vector α, supp(α) ≜ {i : αi ≠ 0} denotes the support set of α
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